Is this for a business?
If for a business and not sure what to do with the domain, I would not touch anything and get some expert assistance to evaluate how things are configured, for expert advise.

This is something that is hard to do via a forum online without plenty of detail.
Why do you need to change the servers name? If you just need an alias to reach it on, you can setup an A record on DNS (IF DNS Server role is deployed and configured on the server)

Or is it the actual domain name you wish to change?
Perhaps you want to add a child domain?

Would need some more information :)
But again, if this is a business domain, and you are not sure on what you are doing, I would be hesitant to make any recommendations which might lead to the domain being at risk of failure.

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I would be tempted to use an alias. Ask the vendor if adding an A record to the DNS and providing him that as FQDN would suffice. It saves work and potential hassles. Vendors should be aware that their clients are not going to re-configure their domain to buy or use a product in many cases. So they should be aware of such workarounds.

There's no such thing as an alias for a domain. A domain has both a DNS name, and a NetBIOS name. The first part of the DNS name, and the NetBIOS name should generally be the same - variations generally just cause confusion with no benefit. So if the DNS name for the domain is "corp.example.com", the NetBIOS name would be "corp".

There is such a thing as an alias DNS record (CNAME) which allows you to use a different name to reach a specific host. For example if the FQDN for the host was "server95485890.corp.example.com" (assume there is an accompanying A record), you could create a CNAME record named "app.corp.example.com" to point to the previously mentioned A record, which would then resolve to the correct IP.

If the feature this vendor wants to implement is really dependent on the DNS name of the domain, then I'd say they're a crap company if they don't support everything that is acceptable by MS as an AD domain.

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